The discrete-time matrices Fd(k) and Gd(k) corresponding to the plant are obtained [26] by the relationships
F(k) = F(kT)T (5-32)
Gd(k) [ e F (kT)tdt]G*(kT) (5-33)
d 0
Where T is again the sample period. By defining Fd(k) and Gd(k) in this manner it is implicitly assumed that the dynamics of the linearized time-varying system (5-13) do not change over any given sample period. Thus, excluding the case when the reference and disturbance signals are constant, equation (5-30) is indeed only an approximation.
Assuming that our discrete-time model is reasonably accurate, the feedback gains Kl(k) and K2(k) are selected to give stability of the discrete-time system (5-30). The actual mechanism for selecting the feedback gains shall not be discussed, however, solving an algebraic Riccati equation would be one approach [24]. In any event there is a stabilizability (controllability) requirement for the discretized system which must be met. In general, the controllability requirement will be met whenever the continuous-time system is controllable [27] so that controllability of the pair given in equation (5-14) is often sufficient.